|
|
|
|
discriminant
|
(Definition)
|
|
|
Let  be any Dedekind domain with field of fractions  . Fix a finite dimensional field extension  and let  denote the integral closure of  in  . For any basis
 of  over  , the determinant
whose entries are the trace of  over all pairs  , is called the discriminant of the basis
 . The ideal in  generated by all discriminants of the form
is called the discriminant ideal of  over  , and denoted
 .
In the special case where  is a free  -module, the discriminant ideal
 is always a principal ideal, generated by any discriminant of the form
 where
 is a basis for  as an  -module. In particular, this situation holds whenever  and  are number fields.
The discriminant is sometimes denoted with
 instead of  . In the context of number fields, one often writes
 for
 where
 and
 are the rings of algebraic integers of  and  . If  or
 is omitted, it is typically assumed to be
 or
 .
The discriminant is so named because it allows one to determine which ideals of  are ramified in  . Specifically, the prime ideals of  that ramify in  are precisely the ones that contain the discriminant ideal
 . In the case
 , a theorem of Minkowski states that any ring of integers  of a number field larger than
 has discriminant strictly smaller than
 itself, and this fact combined with the previous result shows that any number field
 admits at least one ramified prime over
 .
In the special case where ![$ L = K[x]$](http://images.planetmath.org:8080/cache/objects/2895/l2h/img51.png "$ L = K[x]$") is a primitive separable field extension of degree  , the discriminant
 is equal to the polynomial discriminant of the minimal polynomial  of  over ![$ K[X]$](http://images.planetmath.org:8080/cache/objects/2895/l2h/img56.png "$ K[X]$") .
The discriminant of an elliptic curve can be obtained by taking the polynomial discrimiant of its Weierstrass polynomial, and the modular discriminant of a complex lattice equals the discriminant of the elliptic curve represented by the corresponding lattice quotient.
|
"discriminant" is owned by djao. [ full author list (2) | owner history (1) ]
|
|
(view preamble)
Cross-references: quotient, elliptic curve, lattice, complex, modular discriminant, Weierstrass polynomial, polynomial, discriminant of an elliptic curve, minimal polynomial, degree, separable, primitive, prime, strictly, ring of integers, contain, prime ideals, ramified, algebraic integers, rings, number fields, principal ideal, generated by, ideal, trace, determinant, basis, integral closure, field extension, finite dimensional, fix, field of fractions, Dedekind domain
There are 26 references to this entry.
This is version 9 of discriminant, born on 2002-05-05, modified 2006-10-15.
Object id is 2895, canonical name is DiscriminantOfANumberField.
Accessed 7575 times total.
Classification:
| AMS MSC: | 11R29 (Number theory :: Algebraic number theory: global fields :: Class numbers, class groups, discriminants) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
![$\displaystyle \Delta(x_1,\ldots,x_n) := \det[\operatorname{Tr}(x_i x_j)], $](http://images.planetmath.org:8080/cache/objects/2895/l2h/img10.png "$\displaystyle \Delta(x_1,\ldots,x_n) := \det[\operatorname{Tr}(x_i x_j)], $"){kind=small}
